Inverse problems are often encountered in many fields of technology, including engineering, science and mathematics. Solving an inverse problem entails the determination of certain model parameters from a set of observed data or measurements. The mapping of data to model parameters is a result of interactions within a physical system, such as the Earth, atmosphere, gravity, and so forth. For example, in the field of geophysics, geologic model parameters (e.g., conductivity, density, magnetic permeability, porosity, seismic velocity, etc.) are typically identified from some projections that are acquired on the surface of the earth (i.e. observed data) and are related to the model parameters through a forward model.
More precisely, an inverse problem may be formulated as follows:F(m)≈d  (1)where m=(m1, m2, . . . , mn)εM⊂Rn denotes a model parameter set that belongs to a set or family of admissible model parameters M, dεRs represents the observed data, and F(m)=(f1 (m), f2 (m), . . . , fs (m)) represents the forward model that predicts the observed data.
However, as in any inverse problem, there is not one unique solution. Inverse problems are commonly ill-posed; that is, different kinds of model parameter sets can be used to predict the observed data with the same precision. This is due to some degree of uncertainty or inaccuracy that is inherent in most data observations. Uncertainty exists in inverse problems because of a variety of factors, such as poor data calibration, contamination and noise in data measurements, discrete data coverage, approximated physics and conceptualization, discretization of continuous inverse problems, linearization and numerical approximations, model physical assumptions (e.g., isotropy, homogeneity, anisotropy, etc.), limited bandwidth, poor resolution, and so forth.
Uncertainty may be defined as the difference between the one true value that describes a physical quantity at a specific point in time and space and the value reported as a result of a measurement. Estimation of uncertainty involves finding the family M of equivalent model parameter sets m that are consistent with the prior knowledge and fit the observed data dεRs within a prescribed tolerance (tol), as follows:∥F(m)−d∥2<tol  (2)where ∥ ∥2 represents the Euclidean norm, but other norms can also be used.
Quantifying uncertainty is a key aspect in risk management, business analysis, probabilistic forecasting and many other business processes. Model-based applications that incorporate uncertainty evaluation capabilities can provide invaluable guidance in business decision-making, such as whether or not to acquire more data to reduce uncertainty or to proceed with the current path, or whether the potential reward that can be achieved in developing a set of assets is outweighed by the degree of risk and cost involved. For example, in the petrochemical field, the success of finding new oil and gas reserves can be significantly improved by evaluating the uncertainty of developing new leads.
Despite the importance of “measuring” uncertainty to access risk, however, little progress has been made in finding a robust method for estimating inverse problem model uncertainty, especially in parameter spaces with very high dimensionality and/or very costly forward evaluations. For example, Bayesian network-based frameworks have previously been used to estimate uncertainty. However, such frameworks are very inefficient, especially where the number of parameters is very large and/or the forward evaluations are very costly to compute. Sampling within a Bayesian framework incurs very high computational costs because it is performed in parts of the model space with very small likelihood of being consistent with the observed data. As such, these conventional methods are naturally limited to small parameterizations (i.e. low number of parameters) and fast forward solvers.